Week 12.06.2023 – 18.06.2023

TBA

I will introduce the partial Bondi gauge for 4-dimensional spacetimes. This gauge includes the usual Bondi gauge and Newman-Unti gauge and is designed to approach asymptotic boundaries along null rays. The new gauge is defined by three conditions on the metric (g_{rr}=0=g_{rA}) and relaxes the condition on the radial coordinate. I will discuss the solution space and asymptotic symmetries. Most importantly, by relaxing the gauge, we uncover new large symmetries that characterize asymptotically flat spacetimes.

We study Chern-Simons theories at large N with either bosonic or fermionic matter in the fundamental representation. The most fundamental operators in these theories are mesonic line operators, the simplest example being Wilson lines ending on fundamentals. We classify the conformal line operators along an arbitrary smooth path as well as the spectrum of conformal dimensions and transverse spins of their boundary operators at finite 't Hooft coupling. These line operators are shown to satisfy first-order chiral evolution equations, in which a smooth variation of the path is given by a factorized product of two line operators. We argue that this equation, together with the spectrum of boundary operators, are sufficient to determine these operators' expectation values uniquely. We demonstrate this by bootstrapping the two-point function of the displacement operator on a straight line. We show that the line operators in the theory of bosons and the theory of fermions satisfy the same evolution equation and have the same spectrum of boundary operators.

Recently, a relation was introduced connecting codes of various types with the space of abelian (Narain) 2d CFTs. We extend this relation to provide holographic description of code CFTs in terms of abelian Chern-Simons theory in the bulk. For codes over the alphabet Z_p corresponding bulk theory is, schematically, U(1)_p times U(1)_{-p} where p stands for the level. Furthermore, CFT partition function averaged over all code theories for the codes of a given type is holographically given by the Chern-Simons partition function summed over all possible 3d geometries. This provides an explicit and controllable example of holographic correspondence where a finite ensemble of CFTs is dual to "topological/CS gravity" in the bulk. The parameter p controls the size of the ensemble and "how topological" the bulk theory is. Say, for p=1 any given Narain CFT is described holographically in terms of U(1)_1^n times U(1)_{-1}^n Chern-Simons, which does not distinguish between different 3d geometries (and hence can be evaluated on any of them). When p approaches infinity, the ensemble of code theories covers the whole Narain moduli space with the bulk theory becoming "U(1)-gravity" proposed by Maloney-Witten and Afkhami-Jeddi et al.

Recently, a relation was introduced connecting codes of various types with the space of abelian (Narain) 2d CFTs. We extend this relation to provide holographic description of code CFTs in terms of abelian Chern-Simons theory in the bulk. For codes over the alphabet Z_p corresponding bulk theory is, schematically, U(1)_p times U(1)_{-p} where p stands for the level. Furthermore, CFT partition function averaged over all code theories for the codes of a given type is holographically given by the Chern-Simons partition function summed over all possible 3d geometries. This provides an explicit and controllable example of holographic correspondence where a finite ensemble of CFTs is dual to "topological/CS gravity" in the bulk. The parameter p controls the size of the ensemble and "how topological" the bulk theory is. Say, for p=1 any given Narain CFT is described holographically in terms of U(1)_1^n times U(1)_{-1}^n Chern-Simons, which does not distinguish between different 3d geometries (and hence can be evaluated on any of them). When p approaches infinity, the ensemble of code theories covers the whole Narain moduli space with the bulk theory becoming "U(1)-gravity" proposed by Maloney-Witten and Afkhami-Jeddi et al.

General relativity makes spectacular predictions about our world, predictions which have captured the popular imagination more than any other part of physics: gravitational waves, black holes, spacetime singularities. For the mathematician, however, perhaps the most spectacular thing about these predictions is not their exoticness, but, on the contrary, the fact that they all correspond to well-defined mathematical concepts: Indeed, it was precisely through mathematics that these predictions of general relativity were first discovered—originally to much controversy and objection!—and the qualitative mathematical analysis of the Einstein equations remains one of the most powerful ways to understand the great conceptual questions of the theory. This talk will describe some past contributions of mathematics to general relativity and some of the big open conjectures which mathematics hopes to answer in the future.